There are three primary types of arrays, collinear, broadside, and end fire.

Collinear describes two or more things
arranged in a straight line. Extended double Zepps and two half waves in phase
are examples of collinear arrays. A 5/8th wave radiator, which developes gain when placed over an infinite ground plane through ground reflection image, is also a collinear.
The 5/8th wave is collinear with the ground reflection image.

Broadside is used to describe pattern in relationship to spatial area
occupied by the array. While a collinear array is a broadside array, the term
broadside is generally reserved for non-collinear arrangements.

End fire arrays are arrays with elements arranged in a row, running in the direction(s) of maximum
radiation. Yagi antennas, W8JK arrays, and log periodic dipole arrays are all
end fire arrays. They fire in the direction of the boom through the plane
of the elements.

Gain

Before analyzing
stacking (broadside) or collinear antennas, we
really should
understand how antenna systems produce gain. There is a common "Ham-myth" that doubling the number the number of elements doubles
field strength (3dB more gain). This actually isn't true. Doubling the number of
elements, or even doubling array size, can change gain almost any amount. While
there are isolated cases where a size or element number change might result in
3db additional gain, it is through coincidence rather than science. To
understand this, we have to understand what causes "gain".

When a fixed power level is coupled into an antenna, all energy not lost as
heat is radiated as an electromagnetic field. This radiation is unevenly
distributed in space, much like a volume of air in a rubber balloon. The
creation of new or deeper nulls, in angles and directions initially having significant
radiation, forces energy into narrower directions. This increases level in those
areas at the expense of radiation in other direction, much like squeezing a
balloon with our hands will cause a balloon to expand in certain directions at
the expense of others. It is this squeezing of the radiation into a smaller
spatial area that causes a "gain" increase. A newly
created null or a deeper null removes energy from null areas, forcing that energy in
what we hope are more useful or desirable directions. In other words, because applied transmitter power is constant,
energy formerly radiated in areas of significant radiation moves to
enhance field strength in
other directions.
This effect is not from increased element number area or physical array size. The gain increase is actually caused by
redistribution of energy, because of the new, wider, or deeper null areas. When
a cancellation of radiation removes power from areas with significant radiation,
the pattern becomes narrower. This is an increase in directivity. When directivity increases without significantly increasing heat losses,
"gain" has to increase.

Null forming is caused by the relative phase and level relationship between two or more sources of radiation at different points in space around an array. If we arrange elements over space, phasing the elements in a way that forces
a null where there is very little energy, there will not be much pattern directivity or gain change. This
is because the system is attempting to remove energy from an area already lacking significant
energy. The reward is small, because there is little energy to redirect to more
useful angles or directions. An example of this is the
quad element, as height is
varied. When a horizontally polarized full-wave quad element is placed 1/2λ
above ground, or multiples of 1/2λ above ground, quad gain over a dipole is
minimized or can even be negative. This is because the quad element is trying
to force a null in an area that already has a deep null from ground reflection. Conversely, when the
quad element is compared to a dipole , each at 3/4λ high (which produces a strong overhead radiation lobe from ground reflection), or if a quad element is compared to a dipole in free space, quad element gain over a
dipole is maximized. This is because, under the latter conditions, the
dipole had significant energy where radiation from the quad's second current
maximum forces a null.

This effect is
sometimes called pattern
multiplication. Before
antenna modeling
software was
commonly available,
we often used
pattern
multiplication to
estimate patterns
and gain. Pattern multiplication remains a useful tool, helping us visualize why
or how a particular array behaves in a certain way.

We should always consider patterns caused by spacing and
phasing, including earth reflections, when planning optimum spacing. At lower heights optimum
spacing will often vary as element mean height above earth, or null points created by
original cells making up an array, vary. We should never assume a certain spacing always
produces optimum gain, or that
effective
aperture somehow sets optimum broadside or collinear spacing. In fact, doubling an antenna's size almost never doubles antenna gain (3dB).

I've added graphs
from Jasik's Antenna Engineering Handbook throughout this article. These graphs show theoretical
maximum gain of short, lossless, dipole elements
when arranged end-to-end
(Collinear) or
parallel above or next to each
other (stacked
broadside). These graphs represent the maximum possible theoretical or mathematical gain
of array elements having a single current maxima at or near the center, such as a dipole or center fed 5/8th wave element.

Collinear
Gain

First we have the
end-to-end or collinear
element
placement gain.

The "Relative Spacing In Wavelengths" in the graph (fig 5-22) above is the current-maximum
spacing of the
elements, not element "tip" or "center" spacing.

Gain for two collinear 1/2λ (or
shorter) dipole elements peaks with ~0.9λ
(wavelength) spacing.
Radiation is caused by current, and so areas of maximum current are where radiation
primarily occurs. With a
1/2λ dipole in
each element, the
dipole center spacing
below would be 0.9 - .25
-.25 = 0.4λ. The overall array length would
be .9 + .25 + .25 = 1.4λ

With two
dipoles placed end-to-end at the center with almost no end spacing, spacing of current
maximums (S) would be
.25+.25=.5λ. Overall length would be
twice that of a single dipole, or 1λ. Maximum theoretical gain for this spacing is found
on the graph above,
at the crossing of
the vertical 0.5
RELATIVE SPACING IN WAVELENGTHS and
intersection of
"curve 2" (the two-element curve), as just under 2dB over a single
element. Gain
of two half waves in phase with close end-spacing is always less than
2dB over a single element in ANY
collinear antenna using 1/2λ or shorter
elements with very small element end spacing (D).
Despite what is commonly claimed in Ham-myths, we see simply doubling the
number of elements and doubling array length does not guarantee a gain increase of 3dB!

Obtaining 3dB gain by doubling the number of collinear elements from one to
two requires a significant size increase over double-size. A collinear two-element antenna, using half-wave dipole
elements, has
3dB gain over a single dipole when dipole current maxima spacing (S) is .75λ. This is .25λ
D, or 1.25λ overall L in the case of 1/2λ dipole elements, for 3dBd gain. Keep in
mind 3dB is not the maximum obtainable gain for two elements. Maximum
theoretical
gain is about 3.25dBd, and for 1/2λ dipole elements, occurs at about 0.9λ
S current maxima spacing, or 1.45λ L overall length. If the elements were 5/8th wave, the result would actually be less gain! This gain loss occurs from the addition of areas with out-of-phase currents in the 5/8th wave element.

Doubling gain
again (3dB
more gain, for a total of ~6dB), over the two-element case, requires
a minimum of four collinear elements.
In this case array length would be
2.75λ, or 5.5 times the overall
length of the initial dipole.

A four or more element collinear array can produce over 6dBd gain
by a current maxima spacing over 0.75λ. If,
for example, the array
has 0.95λ
current maxima spacings,
a four element array
would have an optimum
gain of 6.7dB. This would result in an overall physical length of 3.35λ.
Again, this does not follow the "double elements is 3dB more gain" myth. Now the
gain is 6.7dBd, it has 4 (or more) times the elements, and has 6.7 times the
length of the initial dipole. To obtain 6.7dB gain over a dipole, the array had
to be 6.7 times the dipole length! To obtain 6 dB over a dipole, the array
length had to be 5.5 times the dipole length.

The more elements the
array has, the further individual
elements must be
spaced for optimum
gain. Doubling physical size while doubling the number of elements does not
double collinear gain.

EXAMPLE OF COLLINEAR

Using EZNEC, we see the gain of a lossless dipole in freespace is 2.14 dBi.

Adding a second collinear element with close end-to-end spacing, which
doubles antenna size, we have:

We now have 3.71-2.14 = 1.57dBd gain. We doubled antenna size, but gain
increased only 1.57 dB. Looking at Jasik's collinear gain graph, we find
very close agreement. Current maximum spacing "S" in the model is 0.5λ, and maximum
theoretical gain predicted in Jasik's graph is about
1.75dB:

We now have 5.37-2.14 = 3.23 dBd gain. This is in close agreement with
Jasik's graphs. 3dB gain increase requires 2.6 times antenna area
increase,
and this is with simple dipoles. Generally, as individual elements or cells (groups) of
elements making up an array become more directive, optimum spacing distance
increases.

Earth Influences on Azimuth-focused Gain

When an antenna with azimuthal or compass-heading directivity increase is
placed above earth, pattern multiplication and gain is not largely affected by
earth. It is still possible to nearly obtain full theoretical gain at any
height. This is because the earth is not trying to force a null where the array
is also trying to create a null.

Broadside Arrays

Broadside array usually describes elements
or cells or elements placed parallel
and one above the
other. The graph below shows OPTIMUM
or maximum possible gain,
not the actual
gain an array might
have. The graph below is for
1/2λ dipole
elements (or shorter) in
freespace. Stacked Yagis
would, in general, require wider
spacing to produce
maximum possible gain, and almost always produce less than the theoretical
maximum gain increase shown below. This is because a Yagi generally has a
significant null off the antenna's forward lobe. The differently located nulls,
or reduced energy level in areas where dipoles normally have significant
radiation, change optimum broadside stacking distance.

Optimum broadside
stacking distance
increases with more
directive elements
or cells. This is
why a pair of multi-element
Yagi's stacked
requires wider
spacing for the same gain increase than a pair
of dipoles, and why
less maximum
stacking gain is
possible with the Yagi than we might
obtain with stacked
dipoles. Think of it
this way, if the
antenna is already
narrow there is less
unwanted energy
available to move to
the main lobe. Again, this shows doubling the number of elements or size does not double the gain.

Here is the
optimum gain graph
for dipoles in
freespace:

We can see
maximum gain occurs
at .675λ stacking
height. The stacking
gain is 4.8db, not
3dB as we often see
claimed. Again the
more elements the
narrower the
pattern, and the
narrower the pattern
the wider spacing
must be between
elements for maximum
gain.

EZNEC Comparison

Here is the freespace EZNEC plot of two stacked (broadside) dipole elements:

Compared to a lossless dipole in freespace, gain is 5.96 - 2.14 = 3.82 dBd.
This agrees with the graph from Jasik.

Earth Influences on Elevation Patterns

The presence of earth influences optimum stacking distance. This is because
the earth is trying to force a null in the same area or areas that elevation
stacking is also influencing. The earth can be considered a second element, and
before computer modeling, this influence was often visualized and calculated by
using fields from an imaginary "image antenna" placed down in the earth. The
"image antenna" was not something that actually existed, but was a tool for
calculating elevation patterns in the presence of earth. If we search old
textbooks and handbooks, the image antenna often appeared.

If we have a lossless dipole at 1/2λ over
perfect earth, we have this basic gain:

For a single lossless dipole, 1/2λ over
perfect earth, gain is 8.4 dBi. Now let's watch when we stack the dipoles at the
same 1/2λ wave stack spacing:

3.) Optimum
stacking distance
for gain is
virtually never 1/2λ, it is
almost always wider.

4.) Optimum
stacking distance
can be very wide for
arrays with multiple sharp-pattern
antennas, or cells, in the array.

5.) Maximum gain
occurs only when a
null is forced in
an area that formerly contained significant energy
levels. If the original element or cell of elements has a null where the
stacking distance tries to force a new null, maximum gain increase is reduced.

7.) Determining optimum stacking height or distance requires a model that
includes earth, as well as feed line and conductor losses.

Feed Systems

The optimum feed system is generally a distributed or branching type of feed
system. There are many articles suggesting feed systems, so I'll only point out
a few places where caution should be applied.

One error is using long lengths of 75-ohm line to co-phase two 50-ohm
elements. An odd-quarter wavelength 75-ohm line transforms impedances because
the line is mismatched, and has standing waves. If the line is lossless with a
perfect 50 0j ohm load, the 75-ohm SWR all along the 75-ohm line is 1.5:1. This
means, at odd-quarter wave distances, line impedance becomes 1.5*75 = 112.5 0j
ohms. Two of those impedances in parallel are 56.25 ohms. Unfortunately, the
required coaxial cable physical length means elements must ether be less than
1/2λ apart, or we must use longer feed lines from
the Q-sections to element centers.

Many articles make the Q-section longer than
λ/4, such as 3λ/4 or 5λ/4. We should be careful doing this, for two reasons:

Losses increase because mismatched sections
are longer

Bandwidth decreases because there are
multiple mismatched sections in series

Consider the case of a line 5λ/4 wavelengths
long. Such a line has five 90-degree sections in series. If frequency changes
2%, it causes a 2% error in each λ/4 section. Errors in each of the five
sections add, and now the total line error is 10%. The Q-section not only has
additional loss, it also has reduced bandwidth.

Using a 50-ohm section from each element to its
respective Q-section, so each Q section only needs to be λ/4 long, will always
increase bandwidth and often decrease losses. It is also not difficult to
implement. My 6-meter Yagi stack uses 50-ohm equal length lines to the
Q-sections, which are only λ/4 long. Length of the 50-ohm sections does not
matter, so long as they are equal, because the 50-ohm sections are matched, and
essentially have a 1:1 VSWR. If I make a velocity factor error, or change
frequency, or have 75-ohm line losses, errors and/or problems are 5 times less!

I'm not forced to have feed lines between the
antennas that can only be changed in multiples of λ/2, such as λ/4, 3λ/4, 5λ/4,
and so on. I can use two equal length 50-ohm feed lines that are any physical length
that reaches, with the only attention to detail the quarter wavelength 75-ohm
Q-sections. This greatly reduces cable cutting errors because the long
lines are matched, and only need to be equal lengths of the same cable stock.

Where do we use
stacking and
collinear gain most
effectively?

We use broadside
stacking and
collinear gain most
effectively in
Curtain Arrays like
the Lazy H antenna.
You can read more
about Curtain
Antenna Arrays on my
Sterba Curtain
Lazy-H antenna page.

Arrays over Earth

Unless we remove energy from an area that had significant energy, antennas
cannot produce gain. If an antenna has a wide area with no radiation at all, and
we designed a system to force a null totally within that same area, there would
be no gain at all. The pattern must be narrowed to increase gain, and it must be
narrowed in a way that does not increase heat losses faster than it concentrates
electromagnetic radiation.

Earth focuses energy in the elevation plane, creating one or more nulls in
elevation pattern. Height above earth, as well as quality of earth, controls the
nulls formed by earth reflections. For somewhat flat earth, nulls formed by the
presence of earth are all in the elevation plane of the pattern. In most cases,
azimuthal beamwidth, or compass directivity, is largely unaffected by antenna
height. For this reason, azimuth pattern multiplication, or gain increase by
focusing in what we consider "compass direction", is largely unaffected by
height above earth or changes in stacking spacing. We saw this above in the
relationships between actual gain with height or vertical spacing of the antenna
or antennas, and gain changes with changes in horizontal spacing or area
occupied by the antenna elements.